1 . . arXiv:1701.00100v1 [math.CA] 31 Dec 2016 , x ( ) ln-1 x ( ) x i , i = -1, R, = 0 ( ). . , . On properties of the coefficients of the complicated and exotic expansions of the solutions of the sixth Painlev´e equation I. V. Goryuchkina It is known, that among the formal solutions of the sixth Painlev´e equation there met series with integer power exponents of the independent variable x with coefficients in form of formal Laurent series (with finite main parts) in log-1 x (complicated expansions), or in x i , where i = -1, R, = 0 (exotic expansions). These coefficients can be computed consecutively. Here we research analytic properties of the series, that are the coefficients of the complicated and exotic expansions of the solutions of the sixth Painlev´e equation. 1. . (y )2 1 1 1 11 1 y= + + -y + + + 2 y y-1 y-x x x-1 y-x y(y - 1)(y - x) x x - 1 x(x - 1) + x2(x - 1)2 a + by2 + c (y - 1)2 + d (y - x)2 , (1) a, b, c, d ­ , x y ­ , y = dy/dx. x = 0, x = 1 x = , , 1) x = z, y = z/w, 2) x = 1/z, y = 1/w, 3) x = 1 - z, y = 1 - w, (. [1]). . x = 0 (1), 2 , . x , ln-1 x(- ) x i ( R, i = -1) ( ) , . [2]. , x , ln-1 x x i ( = 0). , , (1) , . y = k(x) xk, (2) k=0 k(x) ­ k(x) = ckj j, j=0 ckj C, Z, = ln-1 x = x i ( = 0). (3) - , (2), ­ , [2]. , [3] () , . , [3] , ( ) , . -, 3 . [3] . , [3] . , . ( [3], ) , [3], , . , y = 0(x) + u , 0(x) ­ . ( ) , , u = 1(x)x. , (2), . . , k(x) (2) (1), , , . , , . . (. . ), . . k(x) . () 0(x) x i , = 0 (x = 0, 1, ) . [4] , (. . 4 ). , .. k(x) . (. [5], [6]) , ( , ), . . : k(x) (2) (1). , , , , [3] . : [3] , , . 2. . - , . (x) x = 0 r R {}, ln |(x)| lim = r, (4) x0 ln |x| xD D ­ , . (x) x = r R {}, ln |(x)| lim = r, (5) x xD ln |x| 5 D ­ , . (x), r R {} , , r. (2) . g(x, u, u , . . . , u(n)) = 0, (6) (x) xq1uq20(xu )q21 . . . (xnu(n))q2n, (7) (x) ­ - (.. ), q1 C, q21, . . . , q2n Z+, (n)(x) 0, (n)(x) -n. (6) u = = k(x)xk, (8) k=0 k(x) ­ , k R, (kn)(x) 0, k(n)(x) -n, , k+1 > k. (7) (6) - ( ) (q1, q2), q2 = q20 + . . . + q2n. [3] (x) xq1q20(x )q21 . . . (xn(n))q2n q1 + q20. {Qi = (q1i , q2i ), i = 0, . . . , m} ­ - (6), . . , , R ­ (1, 0). Qi, R = ci R. c = min ci. i=0,...,m (7) (6) - , Qi, R = c R, (. [7]), g^(x, u0, . . . , un), g^(x, u0, . . . , un) = 0 (9) 6 ­ . 1. (6) u = , (8), (9) () u = ^, ^ = 0(x). (10) . 0 = 0, (6) u = x0v. (11) G(x, v, v , . . . , v(n)) = 0, (12) v = , = x0 , . , (11) (6) (x) xq1uq20(xu )q21 . . . (xnu(n))q2n (x) xq1+q20vq~20(xv )q~21 . . . (xnv(n))q~2n, (13) q~20, . . . , q~2n Z+, q~20 + . . . + q~2n = q2. , (11) g^(x, u, u . . . , u(n)) xc P0(x, v, v . . . , v(n)), c ­ q1 +q20, P0(x, v, v . . . , v(n)) ­ v, v . . . , v(n) , (6) (13) q1 + q20 > c. (12) xc [P0(x, v0, . . . , vn) + x1P1(x, v0, . . . , vn) + . . . + xtPt(x, v0, . . . , vn)] = 0, vj = xjv(j), P0(x, v0, . . . , vn), . . . , Pt(x, v0, . . . , vn) ­ v0, . . . , vn , 1, . . . , n R+, 1, . . . , n = 0. xc, P0(x, v0, . . . , vn) + x1P1(x, v0, . . . , vn) + . . . + xtPt(x, v0, . . . , vn) = 0, (14) G(x, v0, . . . , vn) = 0. (14) 1 ­ 3 [3] , p G(x, 0, . . . , n) 0 ( 7 , Pj x, 0, . . . , n = 0 p Pj(x, 0, . . . , n) = 0, Pj x, 0, . . . , n ). u = , = ~ k0(x1, . . . , xN ) k0(x) k0=0 (14) u = ^ + w, ^ = ~ 0(x1, . . . , xN ) 0(x). (15) (15) (14), ( , (14) v0, . . . , vn) , P0(x, ^ 0, . . . , ^ n) + P0(x, ^ 0, . v0 . . , ^ n) w0 + . . . + P0(x, ^ 0, . vn . . , ^ n) wn + + . . . + x1P1(x, ^ 0, . . . , ^ n) + · · · = 0, ^ j = xj ^ (j), wj = xj w(j). (16) (16), - , P0(x, ^ 0, . . . , ^ n), p(P0(x, ^ 0, . . . , ^ n)) , - ( P0(x, ^ 0, . . . , ^ n) = 0). (16) . , p(wj) > p(^ j) = 0, P0(x, ^ 0, . . . , ^ n), . . . , Pt(x, ^ 0, . . . , ^ n) v0, . . . , vn , , 2 3 [3], n > · · · > 1 > 0. - (14), (16) , . . P0(x, ^ 0, . . . , ^ n) = 0. - P0(x, v0, . . . , vn) = 0 (11) x c, , g^(x, ^0, . . . , ^n) = 0, ^j = xj ^(j), ^ = x0 ^ . 2 3. . (1) . x2(x - 1)2y(y - 1)(y - x), . 2x2(x - 1)2y(y - 1)(y - x)y - x2(x - 1)2(3y2 - 2xy - 2y + x)y 2+, (17) 8 +2xy(x - 1)(y - 1)(2xy - x2 - y)y - 2y6a + 4a(x + 1)y5- -2 (a + d)x2 + (4a + b + c - d)x + (a - c) y4+ +4x ((a + b + c + d)x + (a + b - c - d)) y3- -2 (b + c)x3 + (a + 4b - c + d)x2 + (b - d)x y2 + 4bx2(x + 1)y - 2bx3 = 0, , (1), . [2]. [2] , 0(x) (2) ­ , = ln-1 x = x i . x , . , , (2) . (17) y = 0(x) + xu. (18) L 0, 0, ¨ 0, U + xM x, 0, 0, ¨ 0, U + H x, 0, 0, ¨ 0 = 0, (19) U = (u0, u1, u2), uj = xju(j), 0 = 0(x), 0 = x d0(x) , dx ¨ 0 = x2 d20(x) dx2 , L 0, 0, ¨ 0, U = 220 (0 - 1) u2 + 20 320 - 30 0 - 30 + 2 0 u1 - (20) 2 6a50 - 10a40 + 4a30 - 4c30 - 30 - 320¨ 0 + 30 20 + 20 + 20¨ 0 - 20 u0, M x, 0, 0, ¨ 0, U H x, 0, 0, ¨ 0 . (19) u = k+1(x) xk. (21) k=0 9 4. . (19) c (21), . (19) u0, u1, u2 , L 0, 0, ¨ 0, U ) xM (x, 0, 0, ¨ 0, U ). x = 0 , L 0, 0, ¨ 0, U . , L , u0, u1 u2 , M ­ . [2] a = c, a, c = 0 (2) k(x), ln-1 x 2(c - a) 0(x) = (c - a)2(ln x + C )2 - . 2a (22) (21) (22) (19) k=1 Lk(0, 0, ¨ 0, k, k, ¨ k) - Nk(0, 0, . . . , ¨ 0, k-1, k-1, ¨ k-1) j = j (x), j = x dj (x) , dx ¨ j = x2 d2j (x) dx2 , Lk 0, 0, ¨ 0, k, k, ¨ k = xk, (23) (24) = L 0, 0, ¨ 0, k, k + (k - 1)k, ¨ k + 2(k - 1) k + (k - 1)(k - 2)k , Nk 0, 0, ¨ 0, . . . , k-1, k-1, ¨ k-1 . , , k(x) - k Lk 0, 0, ¨ 0, k, k, ¨ k = Nk 0, 0, ¨ 0, . . . , k-1, k-1, ¨ k-1 . (25) (25) x , = ( + C)-1 = (ln x + C)-1, C C. (26) , (26) , x dy = -2 dy , dx d x2 d2y dx2 = 4 d2y d2 + (23 + 2) dy . d 10 (-2a2 + a2 - 2ac + c2)4 (-c + a)64 . Lk , ^k(), d^k() d , d2^k() d2 = Nk(), (27) Lk , ^k(), d^k() d , d2^k() d2 = 4P2() d2^k() d2 + 2P1() d^k() d + P0()^k(), ^k() = k(x), P2 P0 ­ , P1 ­ , P2(0), P1(0), P0(0) = 0, Nk() ­ . , = 0 . , a2()2 d2 d2 + d a1() d + a0(), a2(), a1(), a0() ­ , a2() a1(), a0(). , k(x), ln-1(x) c , . , - (21) - (25) ( 0(x) (22)). (27) , .. Lk , ^k(), d^k() d , d2^k() d2 = -8k2^k() + . . . , ­ ( ), . k(x) ­ ln-1 x. a = c = 0 (2) c 1 0(x) = 2a (ln x + , C) C C. (28) , 11 (2) c (22). . . 1. k(x) (k 1) (2) c (22) (28) ln-1 x, . [8] [9]. , . , , . , ( ) k(x) ln-1 x , , . 5. . (21) (19) k(x) xi -4C0(2a - 2c + C1) , (29) x 2c-2a-C1(C12 + 8C1a + 16a2 - 16ac) - 2C1C0 + C02x- 2c-2a-C1 C0, C1 ­ , C0 = 0, 2c-2a-C1 R, 2c - 2a - C1 < 0, = sgn(Im 2c - 2a - C1). (., , [2]) B0 , B1 , B2 , B6 B7 . , ( C0 C1) (2) - (29) (1). , · B0 0(x) = 2c - C3 2a cos2(ln(C2x) C3-2c/2) 1 + sin2(ln(C2x)C3-2c/2) , (30) a = 0, C0 = C32 + 4C3a + 4a2 - 16ac , C2 2c-C3 C1 = C3 - 2a, C32 + 4C3a + 4a2 - 16ac = 0, C3 = 2c, C2 = 0, 2c - C3 R, 2c - C3 < 0, 12 ­ 2at2 + (C3 - 2a)t + 2c - C3 = 0, = sgn(Im 2c - C3); · B1 1 - c/a 0(x) = 1 - , C2x 2c- 2a (31) a = c = 0, C0 = 8 a( c - a) C2, C1 = 4 a( c - a), C2 = 0, Re( 2c - 2a) = 0, = sgn(Im( 2c - 2a)); · B2 1 + c/a 0(x) = 1 -C2x 2c+ 2a , (32) a = c = 0, C0 = -8 a( c + a) C2, C1 = -4 a( c + a), C2 = 0, Re( 2c + 2a) = 0, = sgn(Im( 2c + 2a)); · B6 1 0(x) = 1 + , C2x 2a (33) a = 0, c = 0, C0 = 8aC2, C1 = -4a, C2 = 0, = sgn(Im 2a); · B7 0(x) = 2c - C1 C1 1 sin2(ln(C2x) C1-2c/2) a = 0,C0 = -C1/C2 2c-C1, C2 = 0, 2c - C1 R, = sgn(Im 2c - C1). (34) 2c - C1 < 0, ( C0 C1) (1), (2) (29). , (29) Cxi. 1(x). , . (21) (19), , x. L1 0, 0, ¨ 0, 1, 1, ¨ 1 + N1 0, 0, ¨ 0 = 0, (35) 0, 0, ¨ 0, 1, 1, ¨ 1 (23), L1 0, 0, ¨ 0, 1, 1, ¨ 1 = 220(0 - 1)¨ 1 + 20(320 - 30 0 - 30 + 2 0) 1 13 -2(6a50 - 10a40 + 4a30 - 4c30 + 30 - 320¨ 0 + 30 20 + 20 + 20¨ 0 - 20)1, N1 0, 0, ¨ 0 = 4a50 - 2(4a + b + c - d)40 + 4(a + b - c - d)30 - 630 0 -430¨ 0 + 620 20 - 2(b - d)20 + 620 0 + 220¨ 0 - 20 20 + 20¨ 0 - 20. (29) (35) x = C = Cxi, R, = 0. , , dy dy x = i , dx d x2 d2y dx2 = -22 d2y d2 - ( + dy i) , d . . - . - , . (29), 0, 0, ¨ 0 - Cxi = Cxi 42 0 = A2 + B + , 1 A = 4 + 4(a + c)2 + 4(a - c)2, B = 22 - 4(a - c), 0 = 4 i 3(A2 - 1) -(A2 + B + 1)2 , (36) ¨ 0 = 43(A2(i - )4 + AB(i + )3 + 6A2 (A2 + B + 1)3 - B(i - ) - i - ) . (35) y = 1(x) (A2 + B + 1)6 - 16 4 2 8 p2j j+2 d2y d2 + p1j j+1 dy d + p0j j y + tj j = 0, (37) j=0 p2j, p1j, p0j, tj C, p20 p28 = 0. (37) , y = C1 y1() + C2 y2() + y3(), C1, C2 ­ , y1(), y2(), y3() ­ . , (35) = Cxi ­ (A2 + B + 1)6, 14 , -16 4 2 (37), d2y d2 - (A2 + B 84 + 1)6 20(0 - 1). (38) (38) ­ , 5 = 0, a1, a2, a3, a4 . , = 0 ­ 2, = a1 = a2 ­ 1, a1 a2 ­ (A-42)2+B+1 = 0, a3 a4 ­ 3, a3 a4 ­ A2+B+1 = 0. (37) = 0, , a1, a2, a3, a4. p20 p28 = 0, (37) , . (37) = + aj, , (37) . , , . [10]. , , . , (37) y = CiFi( - aj)( - aj)i lnµi ( - aj)+ i=1,2 + F3( - aj)( - aj)3 lnµ3 ( - aj), (39) F1(), F2(), F3() C{}, 1, 2, 3 C, µ1, µ2, µ3 Z. , . , [11]. (37) . , y3() ­ , y1() y2() ­ , = 0 = . 2. 1(x) (21) (19) (29) Cxi. 15 , 1(x) = y3(Cxi), y3() ­ (37). 2. - (37) , , (39), , - . - 1, (37), - . - -. , ( - ), , ( ), - (- - ) , (37). (37) , - . - (37) ­ [(0, 0), (0, 1), (8, 1), (8, 0)]. - ( ) ( - ), ­ ( - ) ( ). () 0 . 0 - -222 d2y d2 + 2( + dy 2i) d - 2( + i)2y =0 - 222 d2y d2 + 2( + dy 2i) d - 2( + i)2y + ( + i)2 - 1 + 2b - 2d = 0, ­ . , (0, 1), ­ [(0, 1), (0, 0)]. - 16 y = (C1 + C2 ln )1+ i + ( + i)2 + 2( 1 + 2b + i)2 - 2d , (40) ­ y = (C1 + C2 ln )1+ i , C1, C2 ­ . , , (40) , . [7], , (40) y = C1 a1kk + C2 ln a2kk i + a3kk, k=0 k=0 k=0 (41) a1k, a2k, a3k C, a10 = a20 = 1, ( + i)2 + 1 + 2b - 2d a30 = 2( + i)2 , (37). - A48 -222 d2y d2 - 2(3 - dy 2i) d - 2( - i)2y =0 A48 -222 d2y d2 - 2(3 - dy 2i) d - 2( - i)2y - ( - i)2 - 1 + 2b - 2d = 0, ­ . (8, 1), ­ [(8, 1), (8, 0)]. y = (C1 + C2 ln )-1+ i + ( - i)2 + 2( 1 + 2b - i)2 - 2d , (42) ­ y = (C1 + C2 ln )-1+ i , C1, C2 ­ . , 17 , (42) , . (42) y= C1 b1k + C2 ln b2k i + b3k , k k k k=0 k=0 k=0 (43) b1k, b2k, b3k C, b10 = b20 = 1, ( - i)2 + 1 + 2b - 2d b30 = 2( - i)2 , (37). (37) a1 a2, (A - 42)2 + B + 1 = 0. (37) = + aj, j = 1, 2. . , , . 8 P2j j +1 d2y d 2 + P1j j dy d + P0j jy + Tj j = 0, (44) j=0 P2j, P1j, P0j, Tj C, P20 = 0. - (44) ­ (-1, 1), (0, 0), (8, 0), (8, 1). , , = 0, . . , (-1, 1) [(-1, 1), (0, 0)] . -y + y = 0 -y + y = , C, (44), y = C1 = 0 y = , C1 ­ . C1, C2 (C2 C) y = C1 c1kk + C22 c2kk + c3kk, k=0 k=0 k=0 (45) c1k, c2k, c3k C, c10 = c20 = 1, c30 = , (44). 18 (37) a3 a4, A2 +B+ 1 = 0. (37) = + aj, j = 3, 4. . , , . 8 S2j j +3 d2y d 2 + S1j j +2 dy d + S0j j +2y + Kj j = 0, (46) j=0 S2j, S1j, S0j, Kj C, S20 = 0. - ­ (1, 1), (0, 0), (8, 0), (8, 1). - , , - = 0, . . , (1, 1) [(1, 1), (0, 0)] - . 2(y + 3y ) = 0 2(y + 3y ) = , C, (44), y = C1 2 , C1 = 0 y= , C1 ­ . - C1, C2 (C2 C) y = C1 2 d1kk + C2 d2k k + 1 d3k k , k=0 k=0 k=0 (47) d1k, d2k, d3k C, d10 = d20 = 1, d30 = , (44). , 0, , a1, a2, a3, a4 (37) . , y = C1y1() + C2y2() + y3() i i y = C1f1() + C2 ln f2() + f3(), f1(), f2(), f3() ­ , . . y1() = i i C1f1() y2() = C2 ln f2() ­ , - , . 19 y3() = f3() ­ , . , 1(x) = y3(Cxi). 2 k(x) (21) (19) (29). Lk(0, 0, ¨ 0, k, k, ¨ k) - Nk(0, 0, . . . , ¨ 0, k-1, k-1, ¨ k-1) xk, k=1 j, j, ¨ j (23), Lk 0, 0, ¨ 0, k, k, ¨ k - (24), Nk 0, 0, ¨ 0, . . . , k-1, k-1, ¨ k-1 . , (21) (19), k(x), k N Lk(0, 0, ¨ 0, k, k, ¨ k) = Nk(0, 0, . . . , ¨ 0, k-1, k-1, ¨ k-1). (48) , (48) k, , . x = C = Cxi, R, = 0, k(x) = ^k(), k N, (48). , (48) Q2() 2 d2^k() d2 + Q1k() d^k() d + Q0k()^k() = Nk(), (49) Q2() = -22 20(0 - 1), Q1k() = 2 i (i + 2k - 3)Q2() + i Q1(), Q0k() = (k - 2)(k - 1)Q2() + (k - 1)Q1() + Q0(), Q1() = 20(320 - 30 0 - 30 + 2 0), Q0() = -2(6a50-10a40+4a30-4c30+30-320¨ 0+30 20+20+20¨ 0- 20)1, 0, 0, ¨ 0 (36), Nk() ­ . 3. (49) ^k() ( ) . 20 3. k (49). ^k() = rk kjj, (50) j=0 rk Z, kj C. k = 1 1. k = 2. (49) ­ 0 1, . . N2 R2 A2jj, R2 Z, j=0 A2j C. Q2, Q1k, Q2k, , ^2() = r2 2jj (49), j=0 8 B2a2 (2i - )242 + O(3) 2jr2+j = R2 A2j j . j=0 j=0 (51) a = 0, B = 0, 2i - = 0, R. r2, R2, (51) . 2j . k = 3, N3 ­ , ( ) 0 1, ( ) 2. (50) (49) 8 B2a2 (ki - k + )242 + O(3) kjrk+j = Rk Akj j , j=0 j=0 (52) Rk Z, Akj C. k = 2. , ^3() . , , (49) . 2 4. k(x) = ^k() (21) (19) (29) = 0 ( ) . 4 , (49) . 2 21 5. (49) 0, , a1, a2, a3, a4 C, . 5. 0, , a1, a2, a3, a4 C. . (49) k = 2 . , (49) k = 2 , . k = 3, (49) ­ ( ), . , , k. 2 5 , (49) = 0, , a1, a2, a3, a4 C. , (19) (21) (29), Cxi = 0, , a1, a2, a3, a4. , ^k() = a1, a2, a3, a4 C (49). , 3(x) 4(x) (21) (29) (19) Cxi. , . . . k(x) = ^k() ­ = Cxi . 1. Gromak I.V., Laine I., Shimomura S. Painlev´e Differential Equations in the Complex Plain. Berlin, New York: Walter de Gruyter. 2002. 2. .., .. // . 2010. . 71. . 6­118. 3. .. // . 2016. . 17. 2(58). . 64-87 22 4. Guzzetti D. Poles Distribution of PVI transcendents close to a critical point // Physica D. 2012. doi:10.1016/j.physd.2012.02.015. 5. Gontsov, R.R., Goryuchkina, I.V. On the convergence of generalized power series satisfying an algebraic ODE. Asympt. Anal. 2015. 93(4). P. 311­325. 6. Gontsov R., Goryuchkina I. An analytic proof of the Malgrange-Sibuya theorem on the convergence of formal solutions of an ODE. J. Dynam. Control Syst. 2016. V. 22(1). P. 91-100. 7. .. // . 2004. . 59. 3. . 31­80. 8. .. // . ... 2011. 15. 26 . 9. .. , , " . 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